C1 - Coordinate Geometry

If someone could help me with this question, I'd be really grateful! :biggrin:

Ok so -

A circle passes through the points (2,) and (8,0) and has the y axis as a tangent. Find the two possible equations.

Now the answer is (x-5)^2+(y+4)^2=25 or (x-5)^2+(y-4)^2=25 but I'm just not sure how you get it :s-smilie:

I know how you get the x number, you just find the midpoint of 2 and 8 because then you have a vertical bisector of the circle giving the x axis coordinate for the equation but I'm stumped as to how to get the y coordinate! Please could someone explain it to me in terms that aren't too mathsy?

(edited 10 years ago)

Reply 1

brianeverit

9

Original post by Pabblethefish

If someone could help me with this question, I'd be really grateful! :biggrin:

Ok so -

A circle passes through the points (2,) and (8,0) and has the y axis as a tangent. Find the two possible equations.

Now the answer is (x-5)^2+(y+4)^2=25 or (x-5)^2+(y-4)^2.

I know how you get the x number, you just find the midpoint of 2 and 8 because then you have a vertical bisector of the circle giving the x axis coordinate for the equation but I'm stumped as to how to get the y coordinate! Please could someone explain it to me in terms that aren't too mathsy?

Sketch a diagram. Join the point (8,0) to the centre of the circle and use pythagoras.

Reply 2

Shillington

11

First and foremost: draw a picture.

Now, assuming that you said the circle passes through the following points: (2,0) and (8,0) we know the following

The circle has equation

(x-a)^2+(y-b)^2=c^2

We know that this equation has the following pairs of solutions

(2,0)
(8,0)
(0,b) <--- from the picture you drew.

Now solve.

Reply 3

Pabblethefish

OP

1

Original post by Shillington

First and foremost: draw a picture.

Now, assuming that you said the circle passes through the following points: (2,0) and (8,0) we know the following

The circle has equation

(x-a)^2+(y-b)^2=c^2

We know that this equation has the following pairs of solutions

(2,0)
(8,0)
(0,b) <--- from the picture you drew.

Now solve.

I understand (x-5)^2+(y-b)^2=r^2 but I don't understand how you get the y axis point or r :s

Thanks!

Reply 4

Pabblethefish

OP

1

Original post by brianeverit

Sketch a diagram. Join the point (8,0) to the centre of the circle and use pythagoras.

I don't know the centre point though :s-smilie:

(well I do, I looked at the answers) but I don't know how to find the centre point. Am I right in thinking there are two possible circles? One that is above the x axis and one that overlaps the x axis? Hence the two possible equations?

Reply 5

user2020user

16

Original post by Pabblethefish

If someone could help me with this question, I'd be really grateful! :biggrin:

Ok so -

A circle passes through the points (2,) and (8,0) and has the y axis as a tangent. Find the two possible equations.

Original post by Shillington

First and foremost: draw a picture.

Now, assuming that you said the circle passes through the following points: (2,0) and (8,0) we know the following

The circle has equation

(x-a)^2+(y-b)^2=c^2

We know that this equation has the following pairs of solutions

(2,0)
(8,0)
(0,b) <--- from the picture you drew.

Now solve.

The OP didn't specify, so do we actually know that the circle passes through

(2, \ 0) \ ?

(edited 10 years ago)

Reply 6

Pabblethefish

OP

1

Original post by Khallil

The OP didn't specify, so do we actually know that the circle passes through

(2, \ 0) \ ?

Yeah, sorry typo! :tongue:

Reply 7

Pabblethefish

OP

1

This is what I understand so far (sorry that it's inverted!) but I don't really know how you work out the y coordinate for the centre of the circle

Reply 8

brianeverit

9

Original post by Pabblethefish

This is what I understand so far (sorry that it's inverted!) but I don't really know how you work out the y coordinate for the centre of the circle

You know the radius is 5 so you can use Pythagoras to find the y coordinate

Reply 9

Shillington

11

Original post by Pabblethefish

I understand (x-5)^2+(y-b)^2=r^2 but I don't understand how you get the y axis point or r :s

Thanks!

Can you please proof read what you're writing before you post. What does "r :s" mean? The picture you have drawn is correct. You should be able to deduce my final equation from that. Then just plug in the three pairs of points I gave you and write out the resulting equations in terms of a, b and c. Note that b is the same as in the third equation I gave you. This should be clear from the picture.

Reply 10

shubhro

1

Original post by Pabblethefish

If someone could help me with this question, I'd be really grateful! :biggrin:

Ok so -

A circle passes through the points (2,) and (8,0) and has the y axis as a tangent. Find the two possible equations.

Now the answer is (x-5)^2+(y+4)^2=25 or (x-5)^2+(y-4)^2=25 but I'm just not sure how you get it :s-smilie:

I know how you get the x number, you just find the midpoint of 2 and 8 because then you have a vertical bisector of the circle giving the x axis coordinate for the equation but I'm stumped as to how to get the y coordinate! Please could someone explain it to me in terms that aren't too mathsy?

drawing to scale. so, you can now figure out how the two circles would appear on the X Y plane. forming an equation there after should be easy. if you are still stuck, then I will work out the solution for you.